Scharlemann's manifold is standard

Selman Akbulut

Review from Zentralblatt MATH:

In the paper [Duke Math. J. 43, 33-40 (1976; Zbl 0331.57007)], {\it M. Scharlemann} constructed a closed smooth 4-manifold $Q$ by surgery of the product $\Sigma\times S^1$, $\Sigma$ the Poincaré homology 3-sphere, along a loop in $\Sigma\times 1\subset\Sigma\times S^1$ normally generating the fundamental group of $\Sigma$.

Moreover, he constructed a homotopy equivalence $$f: Q\to (S^3 \times S^1)\# (S^2\times S^2),$$ which is not homotopic to a diffeomorphism, and asked the question whether or not $Q$ is diffeomorphic to $(S^3\times S^1)\#(S^2\times S^2)$.

This question has stimulated much research during the past twenty years resulting in some partial answers (see the references of the paper). In the present paper, the author gives a nice proof of the above open question so $Q$ is diffeomorphic to the connected sum $(S^3 \times S^1)\# (S^2\times S^2)$.

Reviewed by A.Cavicchioli

Keywords: smooth 4-manifolds; homology spheres; fake homotopy equivalences; Kirby calculus

Classification (MSC2000): 57N13

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